My crystal ball on this niche subject isn't as good as it used to be but:
Airlines/airports try to run 24/7, including Bank Holidays, including Christmas..including days such as Princess Diana's funeral - I flew that day - , and H&M wedding....I flew that day.... so hopefully there will not be a staff shortage on the 19th so bad that flights are cancelled en-mass.
Whoever came up with the no fly zone rumour needs......well it's MN so can't say it....A complete no fly zone over the UK would knock out easy access to/from the eastern end of a lot of routes over the North Atlantic....cause lots of issues, both political and practical.....so what's known as high level airspace over the UK will stay open.
Low level my best guess, based on my experience over the years, is as I posted upthread:
There will no doubt be some short lived temporary closures of airspace at low altitude over parts of London and possibly Windsor...that might effect some traffic out of Heathrow for an hour or two but not shut the airport down.
No doubt many UK airports might halt ground operations for a few minutes at a critical time in the service as a mark of respect.
Other than that I'd expect it to mostly be "operations normal" but of course I could be completely wrong...